Hi Lynsey, here's my 2 cents on your questions.
Any advice on whether to aim for Sokcho or Gangneung? Gangneung is a bigger city and will have more cake stuff for you. It's actually known for it's coffee roasters. The 2 cities are fairly close and you can easily take a bus to and fro. Both places have craft breweries. I've heard about Craft Root in Seokcho and Budnamu Brewery in Gangneung. But for cheese in Korea, you'd find a very small selection in the smaller cities. Seoul is the place to go for international food.
What's the climbing/teaching scene Iike, is it easy finding partners, easy to access crags without a car: The expat climbing scene has definitely been more vibrant in the past but there are still excited people to climb especially in the bigger cities like Seoul and Busan. Finding English speaking climbers in Gangneung will be harder. But the transportation system in Korea is really good and you can bus/train to places all over the country. There are really good Korean climbers here but the language barrier usually makes it harder to meet them. If you're game to learn the culture and language tho, you'll meet amazing friendly Korean climbers. Teaching is pretty simple. The real hard work is dealing with administration. It will really depend on where you work. I love my job, but others hate there's. Ice climbing is great here and the approaches for many of the climbs are pretty simple. Seoraksan has a lot of ice climbs, but the weekends are really busy and hence can be dangerous...If you can, I highly recommend getting some weekdays off and teach on the weekends. Weekday climbing in Korea is awesome, weekends...well it's a small country with lots of avid climbers who are very experienced or complete unsafe novices.
can you climb rock all year round or does the weather restrict this? If you're willing to travel south like Busan or Jeju Island and suffer a little, you can climb all year round.
Any sport climbing too? Any ski mountaineering? Lots of sport climbing, many cracks are bolted too. Almost zero ski mountaineering as the mountains are really too small or not big enough for that. The highest mountain is on Jeju island at 1950m. The higher mountain on the mainland is in Jirisan National Park at 1915m. But if you're creative, have gear, and up for an adventure you can find something obscure and fun.
Last words...I love living in Daejeon (kinda the centre of the country) but I would love to live in Seokcho for the mountains and ocean. If you're independent, want to experience more of a Korean flavour, are more of an adventure climber rather than a sport/gym climber, then Seokcho is amazing. If not (which is perfectly fine too) then maybe Seoul, Busan or Daegu would be more to your liking.
Hope that helps. If you have any other questions hit me (Eddy Park) up with an email at firstname.lastname@example.org